MacPherson's Conjecture via Hörmander Estimate

In this notes we reprove MacPherson's conjecture on $L^2-(n,q)$-cohomology through Demailly's formulation of Hörmander's Estimate. This approach allows us to weaken the condition of locally semipositivity in Ruppenthal's $L^2$-representation of adjoint bundle. Moreover we prove the MacPherson's conjecture of twisted coefficient bundle under an arbitrary singular hermitian metric. As applications, we study the MacPherson type problem for pluri-canonical bundle.

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